Download Planning Sample Size for Randomized Evaluations

Planning Sample Size for
Randomized Evaluations
Esther Duflo
J-PAL
povertyactionlab.org
Planning Sample Size for
Randomized Evaluations
• General question:
How large does the sample need to be to credibly
detect a given effect size?
• What does “Credibly” mean here?
It means that I can be reasonably sure that the difference
between the group that received the program and the group
that did not is due to the program
• Randomization removes bias, but it does not remove
noise: it works because of the law of large numbers…
how large much large be?
Basic set up
• At the end of an experiment, we will compare the
outcome of interest in the treatment and the
comparison groups.
• We are interested in the difference:
Mean in treatment - Mean in control
= Effect size
• For example: mean of the number of wells in villages
with women vs mean of the number of wells in
villages with men
Estimation
But we do not observe the entire population, just a sample.
In each village of the sample, there is a given number of wells. It is
more or less close to the mean in the population, as a function of
all the other factors that affect the placement of wells.
We estimate the mean by computing the average in the sample

i 1
If we have very few villages, the averages are imprecise. When we
see a difference in sample averages, we do not know whether it
comes from the effect of the treatment or from something else
Estimation
The size of the sample:
• Can we conclude if we have one treated village and one non treated
village?
• Can we conclude if we give textbook to one classroom and not the
other?
• Even though we have a large class size?
• What matter is the effective sample size i.e. the number of treated units
and control units (e.g. class rooms). What is it the unit the case of the
Panchayat?

i 1
The variability in the outcome we try to measure:
• If there are other many non-measured things that explain our outcomes,
it will be harder to say whether the treatment really changed it.
When the outcomes are very
precise
Low Standard Deviation
25
15
mean 50
mean 60
10
5
Number
89
85
81
77
73
69
65
61
57
53
49
45
41
37
33
0
value
Frequency
20
Less Precision
Medium Standard Deviation
9
6
5
mean 50
mean 60
4
3
2
1
Number
89
85
81
77
73
69
65
61
57
53
49
45
41
37
33
0
value
Frequency
8
7
Can we conclude?
High Standard Deviation
8
7
5
mean 50
mean 60
4
3
2
1
Number
89
85
81
77
73
69
65
61
57
53
49
45
41
37
33
lu
e
0
va
Frequency
6
Confidence Intervals
• The estimated effect size (the difference in the sample
averages) is valid only for our sample. Each sample will give a
slightly different answer. How do we use our sample to make
statements about the overall population?
• A 95% confidence interval for an effect size tells us that, for
95% of any samples that we could have drawn from the same
population, the estimated effect would have fallen into this
interval.
• The Standard error (se) of the estimate in the sample captures
both the size of the sample and the variability of the outcome
(it is larger with a small sample and with a variable outcome)
• Rule of thumb: a 95% confidence interval is roughly the effect
plus or minus two standard errors.
Hypothesis testing
Often we are interested in testing the hypothesis that
the effect size is equal to zero (we want to be able to
reject the hypothesis that the program had no effect)
We want to test:
H o : Effect size  0
Against:
H a : Effect size  0
Two types of mistakes
• First type of error : Conclude that there is an effect, when in
fact there are no effect.
The level of your test is the probability that you will falsely
conclude that the program has an effect, when in fact it
does not.
So with a level of 5%, you can be 95% confident in the
validity of your conclusion that the program had an
effect
For policy purpose, you want to be very confident in the answer you
give: the level will be set fairly low.
Common level of a: 5%, 10%, 1%.
Relation with confidence intervals
• If zero does not belong to the 95% confidence interval
of the effect size we measured, then we can be at least
95% sure that the effect size is not zero.
• So the rule of thumb is that if the effect size is more
than twice the standard error, you can conclude with
more than 95% certainty that the program had an
effect
Two types of mistakes
Second type of error: you fail to reject that the program
had no effect, when it fact it does have an effect.
• The Power of a test is the probability that I will be
able to find a significant effect in my experiment
(higher power are better since I am more likely to
have an effect to report)
• Power is a planning tool. It tells me how likely it is
that I find a significant effect for a given sample size
• One minus the power is the probability to be
disappointed….
Calculating Power
• When planning an evaluation, with some preliminary research
we can calculate the minimum sample we need to get to:
–
–
–
–
Test a pre-specified hypothesis: program effect was zero or not zero
For a pre-specified level (e.g. 5%)
Given a pre-specified effect size (what you think the program will do)
To achieve a given power
• A power of 80% tells us that, in 80% of the experiments of this
sample size conducted in this population, if there is indeed an
effect in the population, we will be able to say in our sample
that there is an effect with the level of confidence desired.
• The larger the sample, the larger the power.
Common Power used: 80%, 90%
Ingredients for a power calculation in a
simple study
What we need
Where we get it
Significance level
This is often conventionally set at
5%. The lower it is, the larger the
sample size needed for a give
power
The mean and the variability of the
outcome in the comparison group
-From previous surveys conducted
in similar settings
-The larger the variability is, the
larger the sample for a given power
The effect size that we want to
detect
What is the smallest effect that
should prompt a policy response?
The smaller the effect size we want
to detect, the larger a sample size
we need for a given power
Picking an effect size
• What is the smallest effect that should justify the
program to be adopted:
– Cost of this program vs the benefits it brings
– Cost of this program vs the alternative use of the money
• If the effect is smaller than that, it might as well be
zero: we are not interested in proving that a very
small effect is different from zero
• In contrast, any effect larger than that effect would
justify adopting this program: we want to be able to
distinguish it from zero
• Common danger: picking effect size that are too
optimistic—the sample size may be set too low!
Standardized Effect Sizes
• How large an effect you can detect with a given sample
depends on how variable the outcomes is.
– Example: If all children have very similar learning level without a
program, a very small impact will be easy to detect
• The standard deviation captures the variability in the outcome.
The more variability, the higher the standard deviation is
• The Standardized effect size is the effect size divided by the
standard deviation of the outcome
 d = effect size/St.dev.
• Common effect sizes:
d0.20 (small) d 0.40 (medium) d 0.50 (large)
The Design factors that influence
power
• The level of randomization
• Availability of a Baseline
• Availability of Control Variables, and
Stratification.
• The type of hypothesis that is being tested.
Level of Randomization
Clustered Design
Cluster randomized trials are experiments in which
social units or clusters rather than individuals are
randomly allocated to intervention groups
Examples:
PROGRESA
Village
Gender Reservations
Panchayats
Flipcharts, Deworming
school
Iron supplementation
Family
Reason for adopting cluster
randomization
• Need to minimize or remove contamination
– Example: In the deworming program, schools was chosen
as the unit because worms are contagious
• Basic Feasibility considerations
– Example: The PROGRESA program would not have been
politically feasible if some families were introduced and
not others.
• Only natural choice
– Example: Any education intervention that affect an entire
classroom (e.g. flipcharts, teacher training).
Impact of Clustering
• The outcomes for all the individuals within a unit
may be correlated
–
–
–
–
–
All villagers are exposed to the same weather
All Panchayats share a common history
All students share a schoolmaster
The program affect all students at the same time.
The member of a village interact with each other
• The sample size needs to be adjusted for this
correlation
• The more correlation between the outcomes, the more
we need to adjust the standard errors
Example of group effect multipliers
________________________________
Intraclass
Randomized Group Size_
Correlation
10
50
100
200
0.00
1.00 1.00 1.00
1.00
0.02
1.09 1.41
1.73
2.23
0.05
1.20 1.86
2.44
3.31
0.10
1.38 2.43 3.30
4.57
_________________________________________
Implications
• It is extremely important to randomize an adequate
number of groups.
• Often the number of individual within groups matter
less than the number of groups
• Think that the “law of large number” applies only
when the number of groups that are randomized
increase
• You CANNOT randomize at the level of the district,
with one treated district and one control district!!!!
Availability of a Baseline
• A baseline has three main uses:
– Can check whether control and treatment group were the same or
different before the treatment
– Reduce the sample size needed, but requires that you do a survey
before starting the intervention: typically the evaluation cost go up and
the intervention cost go down
– Can be used to stratify and form subgroups (e.g. balsakhi)
• To compute power with a baseline:
– You need to know the correlation between two subsequent
measurement of the outcome (for example: between consumption
between two years).
– The stronger the correlation, the bigger the gain.
– Very big gains for very persistent outcomes such as tests scores;
Control Variables
 If we have control variables (e.g. village population, block
where the village is located, etc.) we can also control for them
 What matters now for power is ,the residual variation after
controlling for those variables
 If the control variables explain a large part of the variance,
 the precision will increase and the sample size requirement
decreases.
 Warning: control variables must only include variables that are
not INFLUENCED by the treatment: variables that have been
collected BEFORE the intervention.
Stratified Samples
• Stratification: create BLOCKS by value of the control
variables and randomize within each block
• Stratification ensure that treatment and control groups
are balanced in terms of these control variables.
• This reduces variance for two reasons:
– it will reduce the variance of the outcome of interest in
each strata
– the correlation of units within clusters.
• Example: if you stratify by district for an agricultural
extension program
– Agroclimatic factors are controlled for
– The “common district magistrate effect” disappears.
The Design factors that influence
power
• Clustered design
• Availability of a Baseline
• Availability of Control Variables, and
Stratification.
• The type of hypothesis that is being tested.
The Hypothesis that is being tested
• Are you interested in the difference between two
treatments as well as the difference between treatment
and control?
• Are you interested in the interaction between the
treatments?
• Are you interested in testing whether the effect is
different in different subpopulations?
• Does your design involve only partial compliance?
(e.g. encouragement design?)
Power Calculations using the OD
software
• Choose “Power vs number of clusters” in the menu
“clustered randomized trials”
Cluster Size
• Choose cluster size
Choose Significance Level,
Treatment Effect, and correlation
• Pick a : level
– Normally you pick 0.05
• Pick d :
– Can experiment with 0.20
• Pick the intra class correlation (rho)
• You obtain the resulting graph showing power
as a function of sample size.
Power and Sample Size
Conclusions: Power Calculation in
Practice
• Power calculations involve some guess work.
• Some time we do not have the right information to
conduct it very properly
• However, it is important to spend some effort on
them:
– Avoid launching studies that will have no power at all:
waste of time and money
– Devote the appropriate resources to the studies that you
decide to conduct (and not too much).